Locally $\sigma$-compact rectifiable spaces

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\varphi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \varphi =\pi_{1}$ and for every $x\in G$, $\varphi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [3]). Then we prove that every locally $\sigma$-compact rectifiable space with a $bc$-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each $k_{\omega}$-rectifiable space is rectifiable complete.